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Dec 2007

Volume 48, Issue 12, Articles (12xxxx)

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Generalized definition of time delay in scattering theory

Christian Gérard and Rafael Tiedra de Aldecoa

J. Math. Phys. 48, 122101 (2007); http://dx.doi.org/10.1063/1.2816255 (15 pages) | Cited 5 times

Online Publication Date: 3 December 2007

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We advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In two-body scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory.
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03.65.Nk Scattering theory
02.30.-f Function theory, analysis

Multiplicativity of the maximal output 2-norm for depolarized Werner-Holevo channels

S. Michalakis

J. Math. Phys. 48, 122102 (2007); http://dx.doi.org/10.1063/1.2818737 (5 pages) | Cited 2 times

Online Publication Date: 4 December 2007

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We study the multiplicativity of the output 2-norm for depolarized Werner-Holevo channels and show that multiplicativity holds for a product of two identical channels in this class. Moreover, it is shown that the depolarized Werner-Holevo channels do not satisfy the entrywise positivity condition introduced by King and Ruskai (e-print quant-ph∕0909181v2; e-print quant-ph∕0981026v1), which suggests that the main result is nontrivial.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Nonrelativistic Lee model in three dimensional Riemannian manifolds

Fatih Erman and O. Teoman Turgut

J. Math. Phys. 48, 122103 (2007); http://dx.doi.org/10.1063/1.2813026 (20 pages) | Cited 6 times

Online Publication Date: 5 December 2007

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In this work, we construct the nonrelativistic Lee model on some class of three dimensional Riemannian manifolds by following a novel approach introduced by S. G. Rajeev (e-print hep-th∕9902025 ). This approach together with the help of heat kernel allows us to perform the renormalization nonperturbatively and explicitly. For completeness, we show that the ground state energy is bounded from below for different classes of manifolds, using the upper bound estimates on the heat kernel. Finally, we apply a kind of mean field approximation to the model for compact and noncompact manifolds separately and discover that the ground state energy grows linearly with the number of bosons n.
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11.10.Gh Renormalization
11.10.St Bound and unstable states; Bethe-Salpeter equations
02.40.Ky Riemannian geometries

Quasiexactly solvable difference equations

Ryu Sasaki

J. Math. Phys. 48, 122104 (2007); http://dx.doi.org/10.1063/1.2818560 (11 pages) | Cited 4 times

Online Publication Date: 11 December 2007

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Several explicit examples of quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogs of the well-known quasiexactly solvable systems, the harmonic oscillator (with∕without the centrifugal potential) deformed by a sextic potential, and the 1/sin2 x potential deformed by a cos 2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods

Multiparticle quasiexactly solvable difference equations

Satoru Odake and Ryu Sasaki

J. Math. Phys. 48, 122105 (2007); http://dx.doi.org/10.1063/1.2818561 (8 pages) | Cited 4 times

Online Publication Date: 11 December 2007

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Several explicit examples of multiparticle quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable multiparticle Hamiltonians, the Ruijsenaars-Schneider-van Diejen systems. These are difference analogs of the quasiexactly solvable multiparticle systems, the quantum Inozemtsev systems obtained by deforming the well-known exactly solvable Calogero-Sutherland systems. They have a finite number of exactly calculable eigenvalues and eigenfunctions. This paper is a multiparticle extension of the recent paper by one of the authors [ R. Sasaki, J. Math. Phys. 48, 122104 (2007) ] on deriving quasiexactly solvable difference equations of single degree of freedom.
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03.65.-w Quantum mechanics

Stochastic representations of Feynman integration

William Boos

J. Math. Phys. 48, 122106 (2007); http://dx.doi.org/10.1063/1.2812416 (18 pages)

Online Publication Date: 17 December 2007

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For polynomially bounded potentials V such that H = H0+V is essentially self-adjoint on S(Rd)⊆D(H0)∩D(V), this essay offers two reconstructions of Feynman’s sum over histories as the unitary image of a genuine integral with respect to Wiener measure μ of a functional σtx(ω) defined on the space W of Brownian paths ω into momentum space Rd. The first representation, based on Feynman’s original argument, “lifts” σtx(ω) from a “convolutional Trotter product formula” for the Fourier-transformed image matht(p) of the time-evolved wave function φt(x) = exp(−itH)φ(x) in L2(Rd). The second—which varies and extends a construction introduced in a slightly different context by Albeverio and Høegh-Krohn [Mathematical Theory of Feynman Integrals, Springer Lecture Notes in Mathematics Vol. 523 (Springer, New York, 1976) ]—lifts the functional σtx(ω) from a “convolutional Dyson expansion” of the time-evolved momentum-space function matht(p).
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05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
02.50.Ey Stochastic processes
02.30.Uu Integral transforms
02.30.Nw Fourier analysis
05.40.Jc Brownian motion

Parametric-time coherent states for Smorodinsky-Winternitz potentials

Nuri Ünal

J. Math. Phys. 48, 122107 (2007); http://dx.doi.org/10.1063/1.2824498 (20 pages)

Online Publication Date: 19 December 2007

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In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional Kepler problem. In the third case, the system is transformed into four ocillators and the parametric-time coherent states are constructed in two coordinate frames. In the fourth case, the system is transformed into two ocillators with the reflection symmetry and the parametric-time coherent states are constructed in two coordinate frames.
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45.50.Jf Few- and many-body systems
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The orbifolds of permutation type as physical string systems at multiples of c = 26 V. Cyclic permutation orbifolds

M. B. Halpern

J. Math. Phys. 48, 122301 (2007); http://dx.doi.org/10.1063/1.2824499 (28 pages) | Cited 1 time

Online Publication Date: 21 December 2007

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I consider the mathλ,λ prime free-bosonic permutation orbifolds as interacting physical string systems at math = 26λ. As a first step, I introduce twisted tree diagrams which confirm at the interacting level that the physical spectrum of each twisted sector is equivalent to that of an ordinary c = 26 closed string. The untwisted sectors are surprisingly more difficult to understand, and there are subtleties in the sewing of the loops, but I am able to propose provisional forms for the full modular-invariant cosmological constants and one-loop diagrams with insertions.
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11.25.-w Strings and branes
02.40.Pc General topology

Lie fields revisited

Peter Morgan

J. Math. Phys. 48, 122302 (2007); http://dx.doi.org/10.1063/1.2825148 (16 pages) | Cited 1 time

Online Publication Date: 26 December 2007

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A class of interacting classical random fields is constructed using deformed ⋆-algebras of creation and annihilation operators. The fields constructed are classical random field versions of “Lie fields.” A vacuum vector is used to construct linear forms over the algebras, which are conjectured to be states over the algebras. Assuming this conjecture is true, the fields constructed are “quantum random fields” in the sense that they have Poincaré invariant vacua with a fluctuation scale determined by ℏ. A nonlocal particle interpretation of the formalism is shown to be the same as a particle interpretation of a quantum field theory.
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03.70.+k Theory of quantized fields
03.65.Fd Algebraic methods
11.10.-z Field theory
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On the conformal forms of the Robertson-Walker metric

M. Ibison

J. Math. Phys. 48, 122501 (2007); http://dx.doi.org/10.1063/1.2815811 (23 pages) | Cited 2 times

Online Publication Date: 21 December 2007

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All possible transformations from the Robertson-Walker metric to those conformal to the Lorentz-Minkowski form are derived. It is demonstrated that the commonly known family of transformations and associated conformal factors are not exhaustive and that there exists another relatively less well known family of transformations with a different conformal factor in the particular case that K = −1. Simplified conformal factors are derived for the special case of maximally symmetric space-times. The full set of all possible cosmologically compatible conformal forms is presented as a comprehensive table. A product of the analysis is the determination of the set-theoretical relationships between the maximally symmetric space-times, the Robertson-Walker space-times, and functionally more general space-times. The analysis is preceded by a short historical review of the application of conformal metrics to cosmology.
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98.80.Jk Mathematical and relativistic aspects of cosmology
04.20.Gz Spacetime topology, causal structure, spinor structure
95.30.Sf Relativity and gravitation
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Connection between quantum mechanical and classical time evolution of certain dissipative systems via a dynamical invariant

Dieter Schuch

J. Math. Phys. 48, 122701 (2007); http://dx.doi.org/10.1063/1.2823975 (19 pages) | Cited 7 times

Online Publication Date: 20 December 2007

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In a former publication [ D. Schuch and M. Moshinsky, Phys. Rev. A 73, 062111 (2006) ] we have shown that the time evolution of a quantum system with at most quadratic Hamiltonian that can be described by different methods, namely, the time-dependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function is connected via a dynamical invariant, the so-called Ermakov invariant. Since exact invariants of this type also exist for certain effective descriptions of dissipative quantum systems, we will show the relations between these descriptions and the corresponding invariants and will discuss how the results obtained for the conservative systems must be modified in the presence of a linear velocity dependent frictional force.
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03.65.Db Functional analytical methods
03.65.Ge Solutions of wave equations: bound states
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The support of the limit distribution of optimal Riesz energy points on sets of revolution in math3

J. S. Brauchart, D. P. Hardin, and E. B. Saff

J. Math. Phys. 48, 122901 (2007); http://dx.doi.org/10.1063/1.2817823 (24 pages) | Cited 1 time

Online Publication Date: 27 December 2007

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Let A be a compact point set in the right half of the xy plane and Γ(A) the set in math3 obtained by rotating A about the y axis. We investigate the support of the limit distribution of minimal energy point charges on Γ(A) that interact according to the Riesz potential 1/rs, 0<s<1, where r is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on Γ(A) which is supported on the outer boundary of Γ(A). We show that there are sets of revolution Γ(A) such that the support of the equilibrium measure on Γ(A) is not the complete outer boundary, in contrast to the Coulomb case s = 1. However, the support of the limit distribution on the set of revolution Γ(R+A) as R goes to infinity is the full outer boundary for certain sets A, in contrast to the logarithmic case (s = 0).
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02.10.Ab Logic and set theory
02.30.Em Potential theory
02.30.Cj Measure and integration

On convex surfaces with minimal moment of inertia

P. Freitas, R. S. Laugesen, and G. F. Liddell

J. Math. Phys. 48, 122902 (2007); http://dx.doi.org/10.1063/1.2823888 (21 pages) | Cited 2 times

Online Publication Date: 28 December 2007

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We investigate the problem of minimizing the moment of inertia among convex surfaces in math3 having a specified surface area. First, we prove that a minimizing surface exists, and derive a necessary condition holding at points of positive curvature. Then we show that an equilateral triangular prism is the optimal triangular prism, that the cube is the optimal rectangular prism, and that the sphere is (locally) optimal among ellipsoids. Many examples of convex surfaces are examined, among which the lowest moment of inertia is achieved by a truncated tetrahedron. The problem of finding the global minimizing surface remains open. The analogous problem in two dimensions has been solved by Sachs and later by Hall, who showed that the equilateral triangle minimizes the moment of inertia, among all convex curves with given length.
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02.40.Ft Convex sets and geometric inequalities
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An exact fluid model for relativistic electron beams: The many moment case

M. C. Carrisi and S. Pennisi

J. Math. Phys. 48, 123101 (2007); http://dx.doi.org/10.1063/1.2809268 (26 pages) | Cited 2 times

Online Publication Date: 6 December 2007

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An interesting and satisfactory fluid model has been proposed in literature for the description of relativistic electron beams. It was obtained with 14 independent variables by imposing the entropy principle and the relativity principle. Here the case is considered with an arbitrary number of independent variables, still satisfying the above mentioned two principles; these lead to conditions whose general solution is here found. We think that the results satisfy also a certain ordering with respect to a smallness parameter ϵ measuring the dispersion of the velocity about the mean; this ordering generalizes that appearing in literature for the 14 moments case.
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47.10.A- Mathematical formulations
47.40.Nm Shock wave interactions and shock effects
02.30.Jr Partial differential equations

Shock reflection and oblique shock waves

Dening Li

J. Math. Phys. 48, 123102 (2007); http://dx.doi.org/10.1063/1.2821982 (20 pages)

Online Publication Date: 27 December 2007

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The linear stability of steady attached oblique shock wave and pseudosteady regular shock reflection is studied for the nonviscous full Euler system of equations in aerodynamics. A sufficient and necessary condition is obtained for their linear stability under three-dimensional perturbation. The result confirms the sonic point condition in the study of transition point from regular reflection to Mach reflection, in contrast to the von Neumann condition and detachment condition predicted from mathematical constraint.
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47.40.-x Compressible flows; shock waves
47.85.Gj Aerodynamics
47.40.Nm Shock wave interactions and shock effects
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Large deviations and Chernoff bound for certain correlated states on a spin chain

Fumio Hiai, Milán Mosonyi, and Tomohiro Ogawa

J. Math. Phys. 48, 123301 (2007); http://dx.doi.org/10.1063/1.2812417 (19 pages) | Cited 9 times

Online Publication Date: 12 December 2007

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In this paper we extend the results of Lenci and Rey-Bellet [J. Stat. Phys. 119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.
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05.30.-d Quantum statistical mechanics
03.65.Yz Decoherence; open systems; quantum statistical methods

Interacting quantum gases in confined space: Two- and three-dimensional equations of state

Wu-Sheng Dai and Mi Xie

J. Math. Phys. 48, 123302 (2007); http://dx.doi.org/10.1063/1.2821248 (20 pages) | Cited 13 times

Online Publication Date: 18 December 2007

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In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for three-dimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.
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05.30.Jp Boson systems
05.30.Fk Fermion systems and electron gas
05.70.Ce Thermodynamic functions and equations of state
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Determination of small amplitude perturbations for the electric permittivity from partial dynamic boundary measurements

Abdessatar Khelifi

J. Math. Phys. 48, 123501 (2007); http://dx.doi.org/10.1063/1.2817820 (10 pages)

Online Publication Date: 4 December 2007

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The paper deals with reconstruction of small perturbation of the uniform isotropic complex electric permittivity from dynamic measurements of the tangential component of the magnetic field on the boundary (or a part of the boundary) of a domain. The method is based on derived asymptotic inverse Fourier transform of the perturbation in the complex permittivity. Through construction of appropriate test functions by a geometrical control method, we provide a rigorous derivation of the inverse Fourier transform of the perturbations in the electric permittivity as the leading order of an appropriate averaging of the partial dynamic boundary perturbations.
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77.22.Ch Permittivity (dielectric function)
02.30.Zz Inverse problems
03.50.De Classical electromagnetism, Maxwell equations

Coupling constant behavior of eigenvalues of Zakharov-Shabat systems

Martin Klaus and Boris Mityagin

J. Math. Phys. 48, 123502 (2007); http://dx.doi.org/10.1063/1.2815810 (47 pages) | Cited 2 times

Online Publication Date: 5 December 2007

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We consider the eigenvalues of the non-self-adjoint Zakharov-Shabat systems as the coupling constant of the potential is varied. In particular, we are interested in eigenvalue collisions and eigenvalue trajectories in the complex plane. We identify shape features in the potential that are responsible for the occurrence of collisions and we prove asymptotic formulas for large coupling constants that tell us where eigenvalues collide or where they emerge from the continuous spectrum. Some examples are provided which show that the asymptotic methods yield results that compare well with exact numerical computations.
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02.10.Ud Linear algebra
42.65.Tg Optical solitons; nonlinear guided waves
42.81.Dp Propagation, scattering, and losses; solitons

Random matrices, nonbacktracking walks, and orthogonal polynomials

Sasha Sodin

J. Math. Phys. 48, 123503 (2007); http://dx.doi.org/10.1063/1.2819599 (21 pages) | Cited 1 time

Online Publication Date: 6 December 2007

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Several well-known results from the random matrix theory, such as Wigner’s law and the Marchenko-Pastur law, can be interpreted (and proved) in terms of nonbacktracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a role in this approach.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Yn Matrix theory
02.10.Ox Combinatorics; graph theory

Quantum corrections to the semiclassical hydrodynamical model of semiconductors based on the maximum entropy principle

V. Romano

J. Math. Phys. 48, 123504 (2007); http://dx.doi.org/10.1063/1.2819600 (24 pages) | Cited 3 times

Online Publication Date: 7 December 2007

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Quantum corrections to the hydrodynamical model of semiconductors based on the maximum entropy principle are obtained at ℏ2 order with a Chapman-Enskog expansion in the high field approximation, modeling the ℏ2 part of the collision term in a relaxation form. Limiting energy-transport and drift-diffusion models are deduced.
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73.63.-b Electronic transport in nanoscale materials and structures
72.20.Ht High-field and nonlinear effects
66.30.Dn Theory of diffusion and ionic conduction in solids
05.70.Ce Thermodynamic functions and equations of state

Generalized eigenfunctions for Dirac operators near criticality

Peter Pickl

J. Math. Phys. 48, 123505 (2007); http://dx.doi.org/10.1063/1.2809265 (31 pages) | Cited 2 times

Online Publication Date: 7 December 2007

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Critical Dirac operators are those which have eigenfunctions and∕or resonances for E = m. We estimate the behavior of the generalized eigenfunctions of critical Dirac operators under small perturbations of the potential. The estimates are done in the L∞-norm. We show that for small k, the generalized eigenfunctions are in leading order multiples of the respective eigenfunctions and∕or resonances. We also estimate the k-derivatives which are important for estimating decay. The method also applies for other differential operators (for example, Schrödinger operators).
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03.65.Db Functional analytical methods

On the exponential decay of the n-point correlation functions and the analyticity of the pressure

Assane Lo

J. Math. Phys. 48, 123506 (2007); http://dx.doi.org/10.1063/1.2819601 (21 pages) | Cited 2 times

Online Publication Date: 7 December 2007

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The goal of this paper is to provide estimates leading to a direct proof of the exponential decay of the n-point correlation functions for certain unbounded models of Kac type. The methods are based on estimating higher order derivatives of the solution of the Witten Laplacian equation on 1-forms associated with the Hamiltonian of the system. We also provide a formula for the Taylor coefficients of the pressure that is suitable for a direct proof of the analyticity.
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02.30.Jr Partial differential equations

Peel or coat spheres by convolution, repeatedly

Matthias Schmidt and Mike R. Jeffrey

J. Math. Phys. 48, 123507 (2007); http://dx.doi.org/10.1063/1.2816259 (12 pages) | Cited 4 times

Online Publication Date: 10 December 2007

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A convolution transformation is presented that maps the four fundamental measures (Minkowski functionals) of a three-dimensional sphere to those of a sphere with a different radius. It is shown that the set of all these transformations, parametrized by the induced change in radius, forms an Abelian (commutative) group and hence constitutes a flexible framework for the manipulation of spheres. The corresponding one-dimensional case is laid out and the relationship to fundamental measure density functional theory is discussed briefly.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.40.-k Geometry, differential geometry, and topology
02.20.-a Group theory
02.10.Ud Linear algebra

Global well posedness of the relativistic Vlasov-Yukawa system with small data

Seung-Yeal Ha and Ho Lee

J. Math. Phys. 48, 123508 (2007); http://dx.doi.org/10.1063/1.2820988 (19 pages) | Cited 4 times

Online Publication Date: 10 December 2007

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In this paper, we present an existence theory and uniform L1-stability estimate for classical solutions with small data to the Vlasov-Yukawa system. The Vlasov-Yukawa system corresponds to a short-range correction of the Vlasov-Poisson system appearing in plasma physics and astrophysics. For the existence and stability of classical solutions, we crucially use dispersion estimates due to the smallness of data.
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52.25.Dg Plasma kinetic equations
52.25.Fi Transport properties
52.65.Ff Fokker-Planck and Vlasov equation
52.27.Ny Relativistic plasmas
95.35.+d Dark matter (stellar, interstellar, galactic, and cosmological)
95.30.Qd Magnetohydrodynamics and plasmas
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